Computed tomography (CT) systems and methods are widely used, particularly for medical imaging and diagnosis. A CT scan can be performed by positioning a patient on a CT scanner in a space between an X-ray source and X-ray detector, and then taking X-ray projection images through the patient at different angles as the X-ray source and detector are rotated through a scan. The resulting projection data is referred to as a CT sinogram, which represents attenuation through the body as a function of position along one or more axis and as a function of projection angle along another axis. Performing an inverse Radon transform—or any other image reconstruction method—reconstructs an image from the projection data represented in the sinogram.
Various methods can be used to reconstruct CT images from projection data, including filtered back-projection (FBP) and statistical iterative reconstruction (IR) algorithms. Compared to more conventional FBP reconstruction methods, IR methods can provide improved image quality at reduced radiation doses. Various iterative reconstruction (IR) methods exist, such as the algebraic reconstruction technique. For example, one common IR method performs unconstrained (or constrained) optimization to find the argument p that minimizes the expression
                    arg        ⁢                                  ⁢        min            p        ⁢                  ⁢          {                                                              Ap              -              ℓ                                            W          2                +                  β          ⁢                                          ⁢                      U            ⁡                          (              p              )                                          }        ,wherein  is the projection data representing the logarithm of the X-ray intensity of projection images taken at a series of projection angles and p is a reconstructed image of the X-ray attenuation for voxels/volume pixels (or two-dimensional pixels in a two-dimensional reconstructed image) in an image space. For the system matrix A, each matrix value aij (i being a row index and j being a column index) represents an overlap between the volume corresponding to voxel pj and the X-ray trajectories corresponding to projection value i. The data-fidelity term ∥Ap− ∥W2 is minimized when the forward projection A of the reconstructed image p provides a good approximation to all measured projection images . Thus, the data fidelity term is directed to solving the system matrix equation Ap=, which expresses the Radon transform (i.e., projections) of various rays from a source through an object OBJ in the space represented by p to X-ray detectors generating the values of  (e.g., X-ray projections through the three-dimensional object OBJ onto a two-dimensional projection image ).
The notation ∥g|W2 signifies a weighted inner product of the form, gTWg, wherein W is the weight matrix (e.g., expressing a reliability of trustworthiness of the projection data based on a pixel-by-pixel signal-to-noise ratio). In other implementations, the weight matrix W can be replaced by an identity matrix. When the weight matrix W is used in the data fidelity term, the above IR method is referred to as a penalized weighted least squares (PLWS) approach.
The function U(p) is a regularization term, and this term is directed at imposing one or more constraints (e.g., a total variation (TV) minimization constraint) which often have the effect of smoothing or denoising the reconstructed image. The value β is a regularization parameter is a value that weights the relative contributions of the data fidelity term and the regularization term.
Consequently, the choice of the value for the regularization term β typically affects a tradeoff between noise and resolution. In general, increasing the regularization term β reduces the noise, but at the cost of also reducing resolution. The best value for the regularization term β can depend on multiple factors, the primary of which is the application for which the reconstructed image is to be reconstructed. Because IR algorithms can be slow and require significant computational resources, a cut-and-try approach is inefficient (e.g., different values of the regularization term β are used for the IR method until an optimal solution is obtained). Moreover, a single CT scan can be used for more than one clinical application, and, therefore, an ability to adjust the reconstructed image with regards the tradeoff between noise and resolution without repeating the computationally intensive IR algorithm is desirable. Thus, improved methods are desired for rapidly generating and modifying a reconstructed image to optimize a tradeoff between noise and resolution.